The invention relates to a method and mechanism for using and implementing a minimum spanning tree. A minimum spanning tree is the shortest tree that connects a set of points in space. A Euclidean minimum spanning tree is the shortest tree in which the distance between a pair of points is the Euclidean distance.
Numerous advantages in many fields can be achieved by being able to efficiently construct a minimum spanning tree. For example, consider the process for designing an integrated circuit (“IC”). An IC is a small electronic device typically formed from semiconductor material. Each IC contains a large number of electronic components, e.g., transistors, that are wired together to form a self-contained circuit device. The components and wiring on the IC are materialized as a set of geometric shapes that are “placed and routed” on the chip material. During placement, the location and positioning of each geometric shape corresponding to an IC component are identified on the IC layers. During routing, a set of routes are identified to tie together the geometric shapes for the electronic components.
Constructing a minimum spanning tree is particularly useful with respect to the routing step. The minimum spanning tree provides a projection of the shortest connectivity that can be achieved between the components on the IC chip. This projection of the shortest connectivity can be used to develop a general mapping for how the chip should be routed, or even as the initial routing plan for the chip.
To explain approaches for constructing a minimum spanning tree, a useful term to describe is the “cut”, which is a subset of the points. A point-pair crosses the cut if one point is in the cut and the second point is outside the cut. Typically, the cut is a set of vertices that are currently connected in a partially-constructed minimum spanning tree, and the point-pair of interest is a point-pair that crosses the cut that are nearest to each other. The efficiency and speed for determining a minimum spanning tree is highly dependant upon the number of point-pairs to be considered.
Examples of known approaches for constructing a minimum spanning tree are the Prim, Kruskal and Sollin approaches, each of which calls for enumeration for all of the point-pairs involving vertex v that cross the cut formed by T′. In the Kruskal approach, the point-pairs are sorted and are considered in order. In the Prim and Sollin approaches, point-pairs are calculated for every point in the tree against every other point, and the point-pair having the shortest distance is added to the tree. These actions repeat until all points are added to the tree. In effect, all point-pairs must be enumerated at every stage of the process to determine the next point/vertex to add to the tree. A significant drawback to these approaches is that since an advanced IC chip may potentially contain a large number of points (components) to route together, requiring enumeration of every point-pair in the layout to form a minimum spanning tree could be prohibitively expensive.
A preprocessing step can be performed to specify a subset of the point-pairs to consider for the traditional approaches. However, such preprocessing steps are complicated to implement and may consume considerable time and computing resources.